Below the mean values of the most important lunar and solar periods are provided. Good pages that also elaborate on the cycles is here.

Most periods presented, which are shorter or equal to a year's period, have to be averaged over a few periods/cycles (due to variations in orbital periods). Values which are much bigger (like the luni-solar precession) can of course not be seen in one human generation, so one has to do extrapolation in these cases (assuming that it is a circular path).

Of course the orbits of the earth(/sun) and moon are not circular, but for this discussion the approximation of a circle is accurate enough (see Conclusion). The stars are also not fixed (our solar system moves through space), but for this discussion they are assumed fixed.

I am using four reference systems:

- the earth-sun system

Astronomically determined by how the sun is observed from the earth (tropical period: geocentric). Of course there are small changes due to other planets/moon.

- the earth-moon system

Astronomically determined by how the moon is observed from the earth (tropical period: geocentric). Of course there are changes due to other planets/sun.

- the earth-star system

Here the precession of the earth axis (which is due by the solar&lunar forces) becomes evident. This precession is specific to the earth and not for the whole solar system the earth belongs to.

Using this with 2 and 3, one get the sidereal periods (which are purely related to Newton/Kepler laws).

- the time system

All cycles are expressed in the stable time system of Terrestial Time (TT: Years, Days and Hours; defined by standard organization). Some cycles though are also expressed in Solar days (UT, Universal Time), this makes comparing the values easier with some sources (like Thom) and perhaps it is more related to the perception of people in former times.

The relation between TT and UT is DeltaT: TT = UT + DeltaT.

- The input date (
) is normally coming (in the archaeocosmology environment) from monument construction dates that are either carbon dates or guesses. I assume that these dates have an accuracy of around 50 Years. - Under the assumption that carbon dating uses the Julian year (or Year) definition (365.25 Days), there is no error introduced by this conversion.
- The difference between Gregorian calendar date and Julian
calendar date is at 2000 CE (over 500 years since start of Gregorian
calanedar) around 4 Days.

So even if one inputs a Gregorian calendar date (for dates from 1582 CE to now) into the Change date field, the error in the periods is still very small (not visible in the 6 decimal).

- Earth
- Eccentricity cycle

~- ~ and ~ Years (Berger, [1991], page 307)

Eccentricity: ~^{5}

Determined by: Changing eccentricity (e) of the earth's path around the sun.: Duration can more or less be determined by time periods between the glacial maxima during the Quaternary (the '100 kyear' cycle of the Milankovitch cycles)

Seen in the climate

- Perihelion cycle

~, ~ and ~ Years (Loutre, M.F., pers. comm. [2003])

Determined by: Changing earth-sun perihelion angle with regard to the stars.

- Obliquity cycle

~Years

Obliquity: ~Degrees ^{6}

**Determined by:**The angle between the earth-sun plane and equator of the earth.

**Seen in the sky:**Duration can be measured by determining when the sun gets back to the same winter solstice position on the horizon.

**Seen in the climate**: Determines the 41 kyear cycle in the Milankovitch cycles480 BCE by Oenopides (Greek), obliquity has been recognized, its cycle not.

At least recognized in: - Luni-solar precession (precession of equinoxes or
equinoctial precession or Platonic Great Year)

~Years ^{7}

Determined by: The wobble of the earth's axis measured with regard to stars.Duration can be measured by determining when the turning point ('pole' star) of the sky is again at the same star position.

Seen in the sky:

**At least recognized in:**150 BCE by Hipparchus (Greek) [Wilson, 1997, page 13] or earlier.

- Sunspot cycle ~
- Year
(Julian year)

Days

Seen in: Astronomy (John Herschel, 1849 CE)

At least recognized in: 1583 CE by Julian Scaliger (France)

- Tropical year (or solar year)

~ - Day

Hours ~ Solar days

- Sidereal day
^{note}

~Hours

Determined by: The length of time for the vernal equinox to return to the meridian.

Note: I would call this Tropical day, but it is called Sidereal day in Explanatory Supplement to the Astronomical Almanac ([1992], page 48) and consulted people on the HASTRO-L. And in the same spirit; the Explanatory Supplement calls Earth's rotation what I would call Sidereal day.

- Hour

Sec

Determined by: Sec (SI Sec) is measured by atomic time at sea level and standardized (Terreestial Time: TT).

Seen on the horizon:

- Moon

- Lunar nodal cycle ~
- Lunar apse cycle
(or apsis or line of apsides cycle)

~Years ^{8}

- Tropical month ~
- Lunar orbit's
inclination

~Degrees

Determined by: This is the inclination between lunar orbit's plane and the ecliptic plane. This seemed not to have changed over time.

But I propose a more generic term Harmonic sum: , as put forward by Dr. Math on my query (which is building upon the concept of the Harmonic mean).

A derivation of this formula is as follows (lunar draconic month is taken as an example):

The two circular movements (of the nodal cycle (A) and the
tropical
month
(B)) make that the draconic month is shorter than the tropical month.
Lets
assume that the draconic month is x days.

The moon has moved 360*x/'tropical
month' degrees and the nodal cycle
has
moved: 360*x/'nodal cycle' degrees. Both are after a days at
the
same position, so:

1 - x/'tropical month'= x/'nodal cycle'

x= 1/(1/'tropical month' + 1/'nodal cycle')

'draconic month'='nodal cycle'*'tropical month'/('nodal cycle' + 'tropical month')

In general:

X=A*B/(A+B)

or

1/X = 1/B+1/A

Another example, now using an old fashioned
watch, so with a
minute
and hour hand;-)

The minute hand takes 1 hour (B) per revolution, while the hour hand
takes 12 hours (A) to make a revolution.

The question is now how much time does it take when minute and hour
hand are at the same point:

X= 1*12/(12-1)=1h 5.45 min

The above formula works analogous for all of the below mean periods (be aware of possible variations when looking at actual values):

- Earth
- Climatic precession (precession of perihelion,
Berger, [1991],
page 307)

Calculated by solar perihelion cycle (A) and luni-solar precession cycle (B)

Important cycles: ~and ~ Years (=A*B/(A+B)), composite ~ Years ^{9}

Time of perihelion: ~Days

Seen in the sky: Duration when the perihelion and equinox coincidence

Seen in the climate: This is the '21 kyear' cycle in the Milankovitch cycles

At least recognized in: 1618 CE by Kepler (German) [Wilson, 1997] - Anomalistic year

Calculated by solar climatic precession cycle (A) and tropical year (B)

~Days ( =A*B/(A-B)) ~ Solar days

Seen in the sky: Duration can be measured by determining when the solar disc size returns to the same size.

Seen at sea: Duration can be measured by determining when an extra higher/lower tide than normal is happening. - Sidereal year

Calculated by luni-solar precession cycle (A) and tropical year (B)

Seen in the sky: Duration can be measured by determining when the sun arrives at the same star (on the ecliptic).

~Days ( =A*B/(A-B)) ~ Solar days

At least recognized in: 1700 BCE by Babylonians [Wilson, 1997] - Ecliptic year (or eclipse year)

Calculated by lunar nodal cycle (A) and tropical year (B)

~Days ( =A*B/(A+B)) ~ Solar days - Solar
day (LOD:
Length Of Day)

Can be calculated by tropical year (A) and sidereal day (B): (=A*B/(A-B))

~Hours ^{2}

Seen in the sky: The length of time for the Sun to return to the meridian

Seen in planetary programs: The calendar dates one inputs is strongly related to Solar day (the computer program calculates the real Days by incorporating deltaT)

- Earth's rotation
^{note}

Calculated by luni-solar precession cycle (A) and sidereal day (B)

~Hours (=A*B/(A-B))

Seen in the sky: The length of time for the stars to return to the meridian

Seen in the sky: Duration can be measured by determining when the sun is again at the same point against the star.

- Moon
- Synodic
month (or lunar month)

Calculated by tropical year (A) and tropical month (B)

~Days ( =A*B/(A-B)) ~ Solar day

Defined as: Period between the repetition of the same position between earth, sun and object.

Seen in the sky: Duration can be measured by determining when the moon is again in the same lunar phase.

At least recognized in: 1700 BCE by Babylonians [Wilson, 1997] - Day on the moon

Calculated by tropical year (A) and tropical month (B)

~Days ( =A*B/(A-B)) - Anomalistic month

Calculated by lunar apse cycle (A) and tropical month (B)

~Days ( =A*B/(A-B)) ~ Solar day

Seen in the sky: Duration can be measured by determining when the lunar disc size returns to the same size.

Seen at sea: Duration can be measured by determining when a higher/lower tide than normal is happening. - Sidereal
month

Calculated by luni-solar precession cycle (A) and tropical month (B)

~Days ( =A*B/(A-B)) ~ Solar day

Seen in the sky: Duration can be measured by determining when the moon returns to the same star back ground position. - Moon's rotation

Calculated by sidereal month

~Days ~ Solar day

Seen in the sky: The moon keeps facing to the earth with more or less the same surface. - Draconic month (or nodical month)

Calculated by lunar nodal cycle (A) and tropical month (B)

~Days ( =A*B/(A+B)) ~ Solar day

Seen in the sky: Duration can me measured by determining when the moon returns every second time to its node on the ecliptic (the nodes are the points of the object's path that cross the ecliptic plane). - Lunar day (or Diurnal
tide) (which is different then Day
on the moon!)

Calculated by tropical month (A) and sidereal day (B)

~Hours ( =A*B/(A-B))

Seen at sea: Duration can be measured by determining when every second time high/low tide is happening.

The following relations are calculated in the above sections:

Blue: related to earth

Green: related to moon

Yellow: related to stars

Much darker: the reference system's orbits/cycles

Remember that a relation e.g. between Tropical-year/Ecliptic-year/Lunar-nodal-cycle, is defined between the three of them, so one can chose any two to calculate the third one.

I have chosen those orbits, which have the most relations with other orbits/cycles/periods, as my reference orbits/cycles/periods. As said earlier one can taken any other reference scheme (but the relation picture stays the same).

The above only explains what you 'observe in real live'. The fact stands that these observations can be made (how 'simple' they perhaps can be explained)!

Other epoch values can be less accurate, because of
missing proper
time series of periods/cycles/orbits. See the notes.

- Metonic
cycle

Determined by synodic months and tropical years.

An interesting quote from Diodorus Siculus (first century BCE):

"... They also say that the moon, as viewed from this island, appears to be but a little distance from the earth and to have upon it prominences, like those of the earth, which are visible to the eye. The account is also given that the god visits the island every nineteen years, the period in which the return of the stars to the same place in the heavens is accomplished; and for this reason the nineteen-year period is called by the Greeks the year of Meton. ..."

Although not part of the definition, I see also a link between the Metonic cycle and the sidereal month (the same place in the heavens)! This is due to the relation between synodic and sidereal month.

A study on the fitting of these periods in this cycle can be seen on this URL.

~235 synodic months (Days)~ 19 tropical years ( Days)~ 254 (235+19) sidereal months ( Days)

Seen in the sky: Duration can me measured by determining when the moon is again on the same date, in its same phase and at appr. the same star background.

At least recognized in: 1300 BCE by Chinese - Nutation cycle

Determined by the lunar nodal cycle.

~Years

Seen in the sky: an extra deviation from the wobble of the earth axis around the pole star.1728 CE by Bradley

At least recognized in:

- Lunar
major/minor standstill
limit period

Determined by lunar perturbation, lunar parallax, lunar nodal cycle and tropical year

The period (when viewing it along the horizon) between major (or minor) standstill limit is ~18, ~18.5 or ~19 Years (it is not precisely the stated number of years, it can vary with a few days/weeks, sometimes though the 19 solar tropical years is a Metonic cycle and the 18 solar tropical years is sometimes a Saros cycle). On average it is aroundYears (Lunar nodal cycle)

Seen in the sky: Duration can be measured by determining when the moon is at its maximum azimuth. - Saros
cycle

Determined by synodic months, draconic months and anomalistic months

A study on the fitting of these periods in this cycle can be seen on this URL.

~ 223 synodic months (Days)~ 242 draconic months ( Days)~ 239 anomalistic months ( Days) (~ tropical years or ~ ecliptic years)

Seen in the sky: Duration can me measured by determining intervals between eclipses.

At least recognized in: 200 BCE by Chaldaean (Babylonian) - Octaëteris
cycle

Calculated by synodic month and tropical years.

~ 99 synodic months ~tropical years (~ 5 venus synodic years = 7.99 tropical years)

At least recognized in: 500 BCE by Cleostratus of Tenedos (Greece) - Earth rotations per sidereal year

Calculated by sidereal year (A) and earth's rotation (B)

~( =A/B)

- Julian
calendar year

~Days = Solar Days

Seen in: Julian calendar (Julius Caesar, 48 BCE)

At least recognized in: 48 BCE by Sosigenes (Greece) - Gregorian calendar year

~Days = Solar days

Seen in: Gregorian calendar

At least recognized in: 1582 CE by Pope Gregory XIII (Vatican)

- Vernal equinox year

Calculated by tropical year, eccentricity cycle, climatic precession and anomalistic year

**Determined by:**A full revolution (360°) of the sun around the earth related to the mentioned reference point.

At least recognized in: 1582 CE by John Dee (England)

Calculation based on astronomical season length:

Vernal equinox year: ~Days ~ Solar days

Summer solstice year: ~Days

Autumnal equinox year: ~Days

Winter solstice year: ~Days

A comparison between other values can be seen on the astronomical season length page. - Lunar perturbation

Determined by ecliptic year, synodic month and draconic month.

~ 0.5 ecliptic year (~Days) and with maxima at quarter lunar phases and minima at full/new lunar phases

At least recognized in: 920 CE by Aboul-Hassan-Aly-ben-Amajour (Arab) [Thom, 1973] - Length of
astronomical season

Calculated by tropical year, eccentricity cycle, climatic precession and anomalistic year

~ avg. 0.25 tropical year (~Days)
Spring length: - Lunar parallax

Determined by the anomalistic month.

~Days

At least recognized in: 300 BCE by Aristarchus (Greek) [Wilson, 1997] - Many eclipse cycles See also circumstances when solar/lunar eclipse could happen

Summer length:

Autumn length:

Winter length:

- The difference between synodic and sidereal months is (1+0.0808) (determined as above).
- 0.0808 is equal to synodic month/tropical year (1/12.37).
- n tropical years equals to 12.37*n synodic months and thus the number of sidereal months equals to 12.37*n*(1+1/12.37) = 12.37*n + n = 13.37*n
- So the difference between synodic (12.37*n) and sidereal months (13.37*n) is equal to the number (n) of tropical years.

- A synodic month is:

1/'synodic month'= 1/'tropical month' - 1/'tropical year' - Because sidereal month is almost the
same as
tropical month, one can also write it as:

1/'synodic month'= 1/'sidereal month' - 1/'tropical year' - If we
express all periods not in Days but in Years we get:

1/'synodic month'= 1/'sidereal month' - 1 - If we now look over n number of tropical years we get the
formula:

n/'synodic month'= n/'sidereal month' - n - So here you see again that the number of synodic months in n
tropical years is equal to the number of sidereal (or tropical) months
in n
years minus the number of tropical years.

- If one use n=19, when gets the Metonic cycle.

- Year

Solar variations in the vernal equinox year from the mean can be in the order of plus or minus some 15 minutes (1 sigma ~ 5 min).

- Month

The variations in e.g. the synodic month are some 7 hours from the mean, (1 sigma ~ 140 min)

- Day

The variations in the solar day are some 30 sec. from the mean (1 sigma ~ 10 sec) (Stephenson [1997, page 4]).

In some cases the time series for the cycles/periods are derived from well known longitude/angle formula in the following way:

L = p + q*t + r*t^{2} + ...
+ s*t^{n} [deg]

p, q, r, s: arguments of the time series

t: a time length; say of m days (like in Julian ephemeris centuries, where m= 36525)

From this longitude one can determine the cycle length by differentiating the longitude and calculating the time it takes to do one cycle ( 360 degrees):

cycle = 360/(q/m + 2*r/m*t + ... +
n*s/m*t^{n-1})

- The time series (3
^{rd}order) for tropical year comes from this site. - The time series (1
^{st}order and sinus term) for solar day is based on own formula derived mainly from data of Morrison&Stephenson [2004]. - The time series (2
^{nd}order) for lunar nodal cycle comes from (Nautical Almanac Office [1974], page 107)

- The time series (2
^{nd}order) for tropical month comes from (Nautical Almanac Office [1974], page 107) - The time series (2
^{nd}order) for eccentricity comes from (Nautical Almanac Office [1974], page 98). - The time series (7
^{th}order) for obliquity comes from (Bretagnon [1986], page 6). - The time series (6
^{th}order) for luni-solar precession comes from (Bretagnon [1986], page 6). - The time series (2
^{nd}order) for lunar apse cycle comes from (Nautical Almanac Office [1974], page 107). - The time series (2
^{nd}order) for climatic precession cycle comes from (Nautical Almanac Office [1974], page 98).

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Last major content related changes: Feb. 23, 2001